Came back to tell you we just spent our second night with our new lights (no drywall yet) and decided it's much, much brighter than we need. One room is 12x24, 9 foot ceilings with six cans. The other is 12x24, 12 foot ceilings with 9 cans. We opted to rewire and put some on separate switches, rather than remove them. Also, we built on a new addition 4 years ago and put just four cans in each bedroom. Our master is 17x17 and I realized last night that 4 is plenty. So if you still think you want that many, you may want to consider separate switches. Just FYI. Another thing I realize is that I don't think your furniture layout is optimal for watching TV or conversation. Is that how you have it now?

Based on how you're describing, I like the two rows, 3x2. This is just a personal preference thing tho. I wouldn't split them on two dimmers. I can't imagine wanting to set half the can lights on one dim level and half on another. I'd put them on one dimmer, then wire the 2nd dimmer to a couple floor half-outlets or accent lighting. Then you can run the can lights or the other lights (floor lamp) or both. I would also put them slightly farther apart than even spacing wall-light-light-wall. Otherwise it will be overly bright down the center where their beams overlap and not as bright on the fringe. With 13', I'd go something like 3.5' light to wall on each side with 6' between the two rows. That's just what I'd do.

Living room that size you would want at least 9 lights, the family room you can get away with 4-6 depending how much light you would like.

If you are planning an island then try to make the size that will allow you to have 3.6 feet all around and enjoy what you have to work with. The lighting position can be 12 inch centers from cabinet faces to wash color on the cabinets and bring out detail....try for spacing centered on the cabinets around the perimeter of the room. The lights need to be colors that work with the cabinets to make them look good....over the island you might want bright LED to give maximum work light.....lastly puck type lights under cabinets on the perimeter to make the back splash shine. Inspiration or possible rooms that might be what you are trying to envision. Candice Olson has a book about kitchens and bathrooms you can find online or in the library. She discusses recessed lighting in the kitchens I believe and online on her design site. Good luck.